Is there a full and precise formulation of Theorem 1 in the Integrated Gradients paper?
After working on it for a while this is what I got.
Concerning proposition 1 in the paper, a rigorous statement could be the following version of the Gradient Theorem for line integrals:
Proposition 1. (Gradient Theorem for Lipschitz Continuous Functions).
Let $U$ be an open subset of $\mathbb{R}^n$.  If
$F : U \to \mathbb{R}$ is Lipschitz continuous, and
$\gamma : [0,1] \to U$ is a smooth path such that $F$ is
differentiable at $\gamma(t)$ for almost every $t\in [0,1]$, then
$$
    \int_{\gamma} \nabla F(\mathbf{x}) \cdot d\mathbf{x} = F(\gamma(1)) - F(\gamma(0))
    \,.
  $$
I found theorem 1 about symmetry-preserving harder to state rigurously while still fully capturing its intended meaning - this is the best I got:
Theorem 1.
Given $i,j\in \{1,\dots,n\}$, $i\neq j$, real numbers $a&lt;b$, and a
strictly monotonic smooth path $\gamma : [0,1] \to (a,b)^n$
such that $\gamma_i(0)=\gamma_j(0)$ and $\gamma_i(1)=\gamma_j(1)$,
then the following statements are equivalent:
(1) For every $t\in [0,1]$, $\gamma_i(t) = \gamma_j(t)$.
(2) For every function $F : [a,b]^n \to \mathbb{R}$ symmetric
in $x_i$ and $x_j$ and verifying the premises of
proposition 1 with $U=(a,b)^n$ we have
$\int_{\gamma} \frac{\partial F(\mathbf{x})}{\partial x_i} \, dx_i
= \int_{\gamma} \frac{\partial F(\mathbf{x})}{\partial x_j} \,
dx_j$.
Further details and proofs: Symmetry-Preserving Paths in Integrated Gradients